Supersymmetry and the Multi-Instanton Measure

Abstract

We propose explicit formulae for the integration measure on the moduli space of charge-n ADHM multi-instantons in N=1 and N=2 supersymmetric gauge theories. The form of this measure is fixed by its (super)symmetries as well as the physical requirement of clustering in the limit of large spacetime separation between instantons. We test our proposals against known expressions for n < 3. Knowledge of the measure for all n allows us to revisit, and strengthen, earlier N=2 results, chiefly: (1) For any number of flavors N_F, we provide a closed formula for F_n, the n-instanton contribution to the Seiberg-Witten prepotential, as a finite-dimensional collective coordinate integral. This amounts to a solution, in quadratures, of the Seiberg-Witten models, without appeal to electric-magnetic duality. (2) In the conformal case N_F=4, this means reducing to quadratures the previously unknown finite renormalization that relates the microscopic and effective coupling constants, τ_{micro} and τ_{eff}. (3) Similar expressions are given for the 4-derivative/8-fermion term in the gradient expansion of N=2 supersymmetric QCD.